نیرمت یرس تفه م سرد تلاایس کیناکم هتفرشیپ
:لیوحت خیرات 29
/ 2 / 94 یرجم هناسفا رتکد :سرد داتسا
1
1- Obtain the velocity distribution for the modified Stokes second problem consisting of a fluid that is contained between two infinite parallel surfaces separated by a distance ℎ. The upper surface is held fixed, while the lower surface oscillates in its own plane with velocity 𝑈 𝑐𝑜𝑠 𝑛𝑡.
2- The velocity profile in a fluid between two parallel surfaces due to an oscillating pressure gradient was shown to be
A Reynolds number for such a flow may be defined by the following quantity:
For this situation, consider the two asymptotic limits that are defined below:
(a) For 𝑅𝑁<< 1 it might be expected that viscous effects will dominate. Expand the expression for the velocity in this case and obtain an explicit expression for the leading term in the expansion. Interpret the result physically.
(b) For 𝑅𝑁>> 1 it might be expected that viscous effects will be small everywhere expect in the vicinity of the walls. Expand the expression for the velocity for this case, and interpret the result that is obtained.
3- For potential flow due to a line vortex the vorticity is concentrated along the axis of the vortex. Thus the problem to be solved for the decay of a line vortex due to the viscosity of the fluid is as follows:
Here 𝜔(𝑟, 𝑡) is the vorticity, and the maximum circulation around the vortex for any time 𝑡 ≥ 0 is Γ. Look for a similarity solution to this problem of the following from:
Thus obtain expressions for the velocity 𝑢𝜃(𝑅, 𝑡) and the pressure 𝑝(𝑅, 𝑡) in the fluid.
4- The following flow field satisfies the continuity equation everywhere except at 𝑅 = 0,where a singularity exists:
نیرمت یرس تفه م سرد تلاایس کیناکم هتفرشیپ
:لیوحت خیرات 29
/ 2 / 94 یرجم هناسفا رتکد :سرد داتسا
2
Show that this flow field also satisfies the Navier-Stokes equations everywhere except at R=0, and find the pressure distribution in the flow field. Modify the foregoing expressions to the following:
Determine the function f(R) such that the modified expression satisfies the governing equations for a viscous, incompressible fluid and such that the original flow field is recovered for 𝑅 → ∞